#72244404

order	2016
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×C₇×D₄ ⋊ C₂×C₃
Hall	C₃×ΓL(1,𝔽₈) ⋊ H(5,𝔽₂)
Sylow	C₇ ⋊ C₃×#1004

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂³×C₃⋊AGL(1,𝔽₇)
commutator, socle, cosocle	C₂×C₃×C₇	C₂⁴×C₃
Fitting	C₂×C₃×C₇×D₄	C₂×C₃
hypercenter	C₂×D₄	C₃⋊AGL(1,𝔽₇)
hypocenter	C₃×C₇	C₃×H(5,𝔽₂)
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂ ◁ C₂×D₄

718 conjugacy classes of subgroups; only the 204 normal subgroups are shown.

normal subgroups only

This group has 718 conjugacy classes of subgroups — too many to draw. Only its 204 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 168

Polycyclic presentation:

2^4,3,2,3,7|a^2=f,c^a=cf,d^b=df,g^a=g^2,h^a=h^6,h^e=h^4

Conjugacy classes & character table

111 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×#1004, H(5,𝔽₂)×C₃⋊AGL(1,𝔽₇), D₃×#6548⋊C₃, …
maximal normal subgroup of	AGL(1,𝔽₇)×#1004, H(5,𝔽₂)×C₃⋊AGL(1,𝔽₇), D₃×#6548⋊C₃, …
maximal quotient of	#72269479, #72269520

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 2, 3, 7]);;
gap> SetPower(coll, 1, F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.8^6);;
gap> SetConjugate(coll, 8, 5, F.8^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);